Math and stuff

Proposition $(\mathbb{Z}/m\mathbb{Z}) \otimes (\mathbb{Z}/n\mathbb{Z}) = 0$ if $m, n$ are coprime. Solution If $m, n$ are coprime, there must exist integers $p, q$ such that $pm + qn = 1$. Let $a \otimes b \in (\mathbb{Z}/m\mathbb{Z}) \otimes (\mathbb{Z}/n\mathbb{Z})$. \[\begin{align*} a \otimes b &= 1(a \otimes b) \\ &= (pm +...

Proposition Find the length of the following paths: $\gamma(t) = 3t + i, -1 \leq t \leq 1$. $\gamma(t) = i + e^{i\pi t}, 0 \leq t \leq 1$. $\gamma(t) = i\sin(t), -\pi \leq t \leq \pi$. $\gamma(t) = t - ie^{-it}, 0 \leq t \leq 2\pi$. Solution 1 $\int_{-1}^{1}...

Proposition Show that if $f(z) = \frac{az + b}{cz + d}$ is a Mobius transformation then $f^{-1}(z) = \frac{dz - b}{-cz + a}$. Solution \[\begin{align*} f^{-1}(f(z)) &= \frac{d(az + b)/(cz + d) - b}{-c(az + b)/(cz + d) + a} \\ &= \frac{d(az + b) - b(cz + d)}{-c(az +...

Proposition Let $A$ be a ring in which every element $x$ satisfies $x^n = x$ for some $n > 1$ (depending on $x$). Show that every prime ideal in $A$ is maximal. Solution Let $P$ be a prime ideal. Let $x + P \in A / P$ be given. Since...

Proposition Let $x$ be a nilpotent element of a ring $A$. Show that $1 + x$ is a unit of $A$. Deduce that the sum of a nilpotent element and a unit is a unit. Solution Let $x$ be a nilpotent element and $x^n = 0$ for some $n \in...